The network icon has a “Connection information” that should confirm all it does is add DNS and doesn’t replace them.
When editing or creating a connection, the ipv4 tab has Method: “Automatic (DHCP) Addresses only”. This is the key. The DNS (and Search domains) can now be explicitly set. Separate with commas.
Boy, it sure felt good but I edited out all my profanity. 
I’m not a fan of the concoction known as network-manager which this all is. It runs that stupid dnsmasq instance on 127.0.0.1 so DNS is hidden as much as possible.
Hope this helps.